Problem 60 - Precalculus: Mathematical for Calculus

Question

Write the sum using sigma notation. $$2+4+6+\dots+20$$

Step-by-Step Solution

Answer

\( \sum_{n=1}^{10} 2n \)

Step 1: Identify the Pattern

Observe the series 2, 4, 6, ..., 20. Each term is an even number, increasing by 2. We can identify each term in the sequence as part of an arithmetic progression where each number is 2 more than the previous one.

Step 2: Generalize the Term Formula

A term in an arithmetic sequence can be written as \( a_n = a_1 + (n-1) imes d \), where \( a_1 \) is the first term and \( d \) is the common difference. Here, \( a_1 = 2 \) and \( d = 2 \), so \( a_n = 2 + (n-1) imes 2 = 2n \).

Step 3: Determine the Number of Terms

We need to write the sum up to 20 using sigma notation. Find \( n \) such that \( a_n = 20 \). So, \( 2n = 20 \). Solving for \( n \), we get \( n = 10 \). This indicates there are 10 terms in the series.

Step 4: Write the Sigma Notation

Using our term formula and knowing the sequence runs from the first term (\( n=1 \)) to the tenth term (\( n=10 \)), the sum can be expressed in sigma notation as: \[ \sum_{n=1}^{10} 2n \]

Key Concepts

Arithmetic Sequence

An arithmetic sequence is a list of numbers where each subsequent term increases by a constant value, known as the common difference. It's like adding the same amount every time you move from one term to the next.

In the sequence from our exercise, 2, 4, 6, ..., 20, each term increases by 2. This makes it an arithmetic sequence.

Understanding an arithmetic sequence helps you identify the pattern in sequences. You can predict future terms and understand the relationship between them. This foundational concept is crucial in diverse mathematical applications, from simple counting to complex problem-solving.

Arithmetic Progression

Arithmetic progression, also called an arithmetic series, is essentially the sum of a part of an arithmetic sequence. In our case, the series is 2, 4, 6, ..., 20, and when summed up using sigma notation, it forms an arithmetic progression.

This progression helps you manage sequences of numbers that increase or decrease by that fixed common difference. By understanding arithmetic progression, you can efficiently handle real-world scenarios where change happens at a consistent rate. For example, calculating costs that increase by the same amount each period or appraising total accumulated quantities in investment calculations.

Series Sum

The series sum is the result you get when you add up all the terms of a sequence. Using sigma notation, you can succinctly express the sum of a sequence as seen in the exercise, \( \sum_{n=1}^{10} 2n \).

Sigma notation serves as a compact, efficient way to express the sum of sequential numbers. It tells you where to start and end your summing process, thanks to the limits provided (in this case, from 1 to 10). This notation is widely used in mathematics to handle series, making complex calculations more manageable and less prone to error. Understanding how to sum a series is invaluable in both academic settings and practical applications like statistical analysis or data aggregation.

Common Difference

The common difference in an arithmetic sequence is the amount each term increases or decreases from the previous term. This value is denoted as \( d \).

For instance, in our problem, the common difference is 2 because each term is always 2 more than the previous one. This constant value plays a key role in defining the sequence and consequently the arithmetic progression.

Recognizing the common difference is crucial to constructing the general term of the sequence, allowing you to find any term in the sequence without listing all terms. This provides a powerful tool for making quick calculations and evaluations in various mathematical contexts. It's essential for efficient problem-solving, particularly in sequences and series-related problems.